PHY 308 Quantum Mechanics
Course Description:
Quantum mechanics explains the dynamics of atomic, molecular, microscopic, and mesoscopic systems, thus being one of the most critical and relevant theories in physics. In addition, quantum mechanics appears in large-scale scenarios like cosmology or astrophysics. Apart from its relevance in fundamental physics, it has applications in modern technologies like quantum information and quantum computing. However, quantum mechanics is counterintuitive and leads to unexpected behaviors, which makes it one of the most intriguing and beautiful topics to study.
In this course, we will explore the fundamentals of quantum mechanics, bringing a new set of concepts: the probability current, the uncertainty principle, Hilbert spaces and operators, the spin, and the quantization of the action, among others. Indeed, these concepts are key to understanding modern physics in any of its flavors, like nuclear physics, high energy physics, condensed matter physics, quantum chemistry, spectroscopy, chemical physics, and atomic, molecular and optical (AMO) physics.
A detailed index of the course is given below:
Chapter 1. Essential classical mechanics and mathematical methods
This chapter reviews the most relevant classical mechanics concepts required in quantum mechanics: Lagrangian, Hamiltonian, action, and abbreviated action. Similarly, we will review mathematical methods for solving ordinary and partial differential equations and essential complex variable techniques.
Chapter 2. A first approach to wave-mechanics
This chapter will introduce Bohr's atomic model, which motivated the application of quantum mechanics to atoms and molecules. This model will help us understand the important link between the quantum realm and classical mechanics. Then, we will briefly introduce wave mechanics, i.e., the study of wave phenomena and how they help introduce quantum mechanics.
Chapter 3. The formal structure of quantum mechanics
This chapter will study the basics of the mathematics behind quantum mechanics. In particular, we will introduce the concept of Hilbert space, operators, eigenfunctions, and eigenvalues, along with commutation relations and their properties.
Chapter 4. The postulates of quantum mechanics
This chapter will present the six postulates of quantum mechanics and how they give rise to the whole quantum world.
Chapter 5. 1-dimensional quantum mechanics
This chapter will present a study on quantum mechanical scenarios in 1-D. For instance, we will cover a particle in a box, bound states, scattering states, and barriers, introducing the concept of reflection and transmission coefficients for particles.
Chapter 6. The quantum harmonic oscillator
This chapter will be dedicated to studying the harmonic oscillator from a quantum mechanical perspective due to its importance in different fields of physics. In class, we will use two different formalisms to solve the quantum harmonic oscillator: the algebraic one, based on the introduction of creation and annihilation operators, and the other solving directly the Schrödinger equation.
Chapter 7. Quantum mechanics in 3-D
In this chapter, we will solve Schrödinger’s equation for 3-D systems: 3D harmonic oscillator and a particle in a 3D box. Next, we will derive the Schrödinger equation for systems with spherical symmetry, i.e., central potentials, leading to the concept of angular momentum.
Chapter 8. Quantum angular momentum theory
This chapter will cover the basics of the quantum theory of angular momentum. Specifically, we will introduce the angular momentum operator and calculate its spectra leading to spherical harmonics—one of physics's most relevant special functions.
Chapter 9. The hydrogen atom
At this point in the course, we count on all the tools and concepts necessary to face any realistic quantum mechanical problem. This chapter will deal with the hydrogen atom, the cornerstone of modern atomic and molecular physics. The hydrogen atom serves as an arena for studying the basics of the electronic structure of atoms and molecules and understanding the shell structure of atoms.
Chapter 10. The spin
This chapter introduces the concept of spin, a quantal magnitude without a classical analog. To demonstrate this, we will introduce spin via the famous Stern-Gerlach experiment. In addition, we will study its commutation relations and matrix representations, as well as time-dependent problems involving spin in external fields and its applications to modern technology like MRI.
Chapter 11. Symmetries in quantum mechanics
This chapter will present the difference between bosons and fermions and the indistinguishability of particles characteristic of quantum mechanics. In particular, we will discuss the N-body problem from a quantal perspective.
Recommended textbooks (optional):
Using different textbooks to cover various aspects of the same topic is preferable. This way, students can observe different perspectives and pedagogical styles to develop their intuition further.
- S. Gasiorowicz, “Quantum Physics” (John Wiley, 1996)
- D. Griffins, “An introduction to Quantum Mechanics” (Cambridge University Press, 1995)
- D. A. Fleisch, “A Student´s Guide to the Schrödinger Equation” (Cambridge University Press, 2018).
- L. D. Landau and E. M. Lifshitz “Quantum mechanics” (Butterworth-Heinemann, 2003)